TENSOR PRODUCT OF ENDO-PERMUTATION MODULES
Ahmad M. Alghamdi And Makkiah S. Makki
Abstract. In this paper, we study some properties of the exterior tensor prod- uct on the category of modules. For this, we prove that the exterior tensor product of two permutation, endo-permutation, endo-trivial and endo-monomial modules are still permutation, endo-permutation endo- trivial and endo-monomial modules respectively. Also, we prove that the cap of an exterior tensor product of two modules equal the exterior tensor product of their caps. Also, we prove that the exterior tensor product of two Dade algebras is a Dade algebra.
2000Mathematics Subject Classification: 20C20; 16S50.
Keywords: Tensor product, endo-permutation modules, Dade algebra.
1. Introduction
The concept of Endo-permutation modules for ap-groupGfor some prime num- ber p is invented by E. Dade in two papers [8, 9]. It is a generalization of the permutation modules which is due to Green [10, 11]. E. Dade studied many prop- erties of this class of modules and classified endo-permutation modules for abelian p-groups. The construction in [2] by J. Alperin is an important step for studying endo-permutation modules. However, the classification of endo-permutation mod- ules over non- abelian p-groups took a long time to achieve. It has been completed by many authors and ended in 2004. See [15] for more details.
The importance of the class of endo-permutation modules can be seen when one studies the sources of simple G-modules in the case thatGis ap-solvable group [14, Chapter 5] as well as when one studies the equivalence between blocks.
Another class of modules which has been introduced and studied by many au- thors is called endo-trivial modules [3, 9]. It is well known that each endo-trivial module is an endo-permutation module. This notion has been used as a tool to investigate and classify endo-permutation modules. The importance of the class of endo-trivial modules can be seen when one studies the equivalence of the stable category of a block [4]. Jon Carlson and Jacques Thevenaz classified this class of modules [5]. For a recent work on endo-trivial modules, the reader can consult the papers [6, 7].
Our purpose in this paper is to study the exterior tensor product of the category of these modules. Our motivation stems from the fact that the tensor product is an operation in mathematics which enables us to construct new objects from others.
For the coefficients, we consider for a prime number p a p-modular system (K, R, F). This means that R is a complete discrete valuation ring, K is the quo- tient field of R of characteristic zero and F is the field of residue of R which is an algebraically closed of characteristicp. For notation, we shall writeOto denoteRor F. We assume that all modules are free overO and finitely generatedOG-modules where Gis an arbitrary finite group.
Our paper is organized as a sequence of sections each of which is concerned respectively to the tensor product of permutation modules, endo-permutation mod- ules, endo-trivial modules, Dade’s algebras and endo-monomial modules. We use the notation ⊗to mean ⊗O.
2.Tensor product of permutation modules
LetG be a finite group and M be an OG-module. We start with the following definition which is due to Green in [11].
Definition 2.1. An OG-module M is called an OG-permutation module if it has an invariant O-basis X under the action ofG. In this case we writeM =OX.
For any two finite groups G1 and G2, the exterior tensor product of two OG1 and OG2-modules can be defined as follows:
Definition 2.2. If Mi is anOGi-module fori= 1,2 the exterior tensor product of Mi is an O[G1×G2]-module with underlying O-module M1⊗M2 and the action of G1×G2 is given by: (g1, g2).(m1⊗m2) =g1m1⊗g2m2.
Since all modules we are dealing with are finitely generated, any module M has a finite subset, say X = {xi :i= 1,2, ..., n} such that any element m∈ M can be written uniquely as m= Σni=1αixi withαi∈ O.
We call a finite set X a G-set if it is endowed with an action of G on it. The following lemma explains the exterior tensor product of two permutation modules.
The proof is easy but, we include it for completeness.
Lemma 2.3. If Mi is a permutation OGi-module, wherei= 1,2 thenM1⊗M2 is a permutation O[G1×G2]-module.
Proof. Since each Mi is a permutation OGi-module for i = 1,2 then from Definition 2.1, each Mi can be written in the the form Mi = kXi for i = 1,2 where X1 and X2 are the invariant basis for M1 and M2 respectively. Now since M1⊗M2 is an O[G1×G2]-module such thatM1⊗M2 =OX1⊗ OX2.Therefore,
M1 ⊗M2 = O[X1×X2]. This means that the module M1 ⊗M2 has an invariant basis X1×X2. So,M1⊗M2 is a permutation O[G1×G2]-module.
3.Tensor product of endo-permutation modules
LetGbe ap-group for some prime numberp. For anyOG-moduleM we denote by EndO(M) the endomorphism algebra of M. This algebra can be endowed with an OG-module structure coming from the action ofGby conjugation; that is if g∈G and ϕ∈EndO(M), then gϕ(m) =g.ϕ(g−1.m) for all m∈M. Dade [8] defined the concept of an endo-permutation module as:
Definition 3.1. An OG-module M is called an endo-permutation module if EndO(M) is a permutation OG-module.
It is clear that each endo-permutation module is a permutation module.
Now we need the following two lemmas:
Lemma 3.2. For anyOG-moduleM, we have EndO(M)∼=M⊗M∗ as anOG- module, where M∗ =Hom(M,O) is the dual OG-module and the tensor product is over O.
From the previous lemma, we see thatM is an endo-permutationOG-module if and only if M⊗M∗ is a permutation OG-module.
Lemma 3.3. Let Gi be a finite group and Mi an OGi-module, where i= 1,2.
Then EndO(M1⊗M2)∼=EndO(M1)⊗EndO(M2).
We shall prove the main theorem in this section:
Theorem 3.4. Let Gi be a finite group and Mi an endo-permutation OGi- module, where i= 1,2. Then M1⊗M2 is an endo-permutationO[G1×G2]-module.
Proof. Since each Mi is an endo-permutation OGi-module for i = 1,2 then from Definition 3.1 we have that EndO(Mi) is a permutation OGi-module. Now using Lemma 3.2 and Lemma 3.3 we see that the tensor module M1 ⊗M2 is an endo-permutation module as EndO(M1⊗M2) is a permutation module.
Recall that an indecomposable OG-module M is called H-projective for some subgroup H of G if M is a direct summand of the induced OG-module IndGH(N) for some OH-moduleN. A minimal subgroup of the collection of all subgroups of G for whichM isH-projective is called a vertex of M. It turns out that the vertex is a p-subgroup ofG. We write Vertex(M) for the vertex group ofM.
We are interested in indecomposable endo-permutationOG-modules with max- imal vertex G, because they are the ones that appear in representation theory. All endo-permutation modules can be described from the knowledge of the indecompos- able ones having maximal vertex (see [14] for more details).
Definition 3.5. An endo-permutation OG-module M is said to be capped if it has at least one indecomposable direct summand with vertexG. Such indecomposable direct summand is called the cap of M and is denoted by Mc.
In particular, if M is indecomposable, then M is capped if and only if it has vertex G.
We shall use the following result which is due to B. K¨ulshammer [13] for dealing with the tensor product of capped module.
Lemma 3.6. Suppose that Mi is an indecomposable OGi-module; i=1, 2 with vertex (Mi) = Vi. Then M1 ⊗M2 is an indecomposable O[G1×G2]-module with vertex (M1⊗M2) =V1×V2.
In the following theorem we shall prove a result about the relationship between the tensor product of the cap of two endo-permutation modules and the cap of their tensor product.
Theorem 3.7. Let Mi be an endo-permutation OGi-module for i= 1,2. Sup- pose that Mic is the cap of Mi. Then M1c⊗M2c = (M1⊗M2)c where (M1⊗M2)c is the cap of the endo-permutation O[G1×G2]-moduleM1⊗M2.
Proof. Since Mic is the cap of Mi that is an indecomposable direct summand of Mi with vertex Gi, for i= 1,2 Lemma 3.6 yields thatM1c⊗M2c is an indecom- posable O[G1×G2]-module with vertex G1×G2. However,M1⊗M2 has a unique indecomposable direct summand with vertex G1×G2. So, the result is complete.
4.The tensor product of endo-trivial modules
We continue to assume thatGis ap-group for some prime numberpandOis either an algebraically closed field of characteristicp or a complete discrete valuation ring of characteristic zero.
Definition 4.1. An OG-moduleM is called endo-trivial if there exists a projec- tive OG-module F such that EndO(M)∼=O ⊕F as an OG-module.
Theorem 4.2. If Mi is an endo-trivialOGi-modules, where i= 1,2 thenM1⊗ M2 is an endo-trivial O[G1×G2]-module.
Proof. SinceMiis an endo-trivialOGi-module, there is a projectiveOGi-module Fi such thatEndO(Mi) ∼=OL
Fi fori= 1,2. By Lemma 3.3, EndO(M1⊗M2)∼= EndO(M1)⊗EndO(M2).So,
EndO(M1⊗M2)∼= (O ⊕F1)⊗(O ⊕F2).
Therefore,
EndO(M1⊗M2)∼= (O ⊗ O)⊕(F1⊗ O)⊕(O ⊗F2)⊕(F1⊗F2).
However, as O[G1 ×G2]-isomorphism, we have O ⊗ O ∼= O, F1 ⊗ O ∼= F1 and O ⊗ F2 ∼= F2. Also F0 := F1 ⊗F2 is a projective O[G1 ×G2]-module. Hence, EndO(M1 ⊗M2) ∼= O ⊕F1⊕F2⊕F0. It follows that EndO(M1⊗M2) ∼=O ⊕F, where F :=F1⊕F2⊕F0 is a projective O[G1×G2]-module. By the definition, this means that M1⊗M2 is an endo-trivial O[G1×G2]-module.
5.The tensor product of Dade algebras
A G-algebra over O is an O-algebra endowed with an action of the group G by algebra automorphisms such that Ψ(g)(a) = ga for a ∈ A,Ψ ∈ aut(A). For any G-algebra A we define the set of G-fixed elements of A byAG={a∈A:
ga=a ∀ g∈G}.
LetH be a subgroup ofG. We consider the relative trace mapT rHG :AH →AG such thatT rHG(a) =P
t∈T ta,whereT is a set of representatives of the left cosets of H inG. It is clear that the image ofT rHG is an ideal ofAG. For technical reason and as we use p-modular system, we shall write the sum of the image of the trace map and the ideal℘AH asAGH, where℘ is the unique maximal ideal inO. We define the Brauer quotient asA(H) :=AH/P
K<HAHK.
Definition 5.1. Dade G-algebra A is an O-simple permutation G-algebra such that A(G)6= 0.
Suppose thatG1 andG2 are two finite groups and letAi;i= 1,2 be aGi-algebra over O, the tensor algebra A1N
A2 can be regarded as a G1×G2-algebra by the action (g1,g2)(a1⊗a2) =g1a1⊗g2a2,for all (g1, g2)∈G1×G2, ai ∈Ai.
Now to introduce the main result in this section we recall the following lemma about the H1×H2-fixed elements of A1N
A2 as well as the image of the tensor of the relative trace maps. For more details, see [1, Lemma 2.1 and Lemma 2.3].
Lemma 5.2. Assume that Ki≤Hi ≤Gi for i= 1,2. Then (A1⊗A2)H1×H2 ∼=AH11 ⊗AH22, and
(A1⊗A2)HK1×H2
1×K2
∼=A1HK1
1⊗A2HK2
2.
Theorem 5.3. The tensor product of any two Dade algebras is a Dade algebra.
Proof. Suppose that Ai is a Dade Gi-algebra for i= 1,2. This means that each Ai is anO-simple permutation Gi-algebra such thatAi(Gi)6= 0.It is clear that the tensor product A1⊗A2 is anO-simple permutationG1×G2-algebra.
Now let us assume that A1⊗A2(G1×G2) = 0. Then we have (A1⊗A2)G1×G2 = X
H1×H2≤G1×G2
[A1⊗A2]GH1×G2
1×H2.
Using Lemma 5.2, we see that
AG11 ⊗AG22 = ( X
H1≤G1
A1GH1
1)⊗( X
H2≤G2
A2GH2
2).
It follows that, AGi i = P
Hi≤GiAiGHi
i. This means that Ai(Gi) = 0, which is a contradiction. Hence, the result is the tensor product of any two Dade algebras is a Dade algebra.
6.The tensor product of the monomial modules
The notion of monomial representations arises for induction from linear repre- sentations. Let G be a finite group and H a subgroup of G. Consider the linear characters ofHwhich are the homomorphisms fromHto the multiplicative group of O. LetN be anOH-module. Assume that Hacts on N via the linearO-characters of H. We say thatN is an OH-module of O-rank one if N is isomorphic to O as O-module and H acts as follows: for h∈H and forn∈N we have h·n=λhnfor someλh ∈ O. Note that the action is well defined as we have a homomorphism from H to the group of units of O. For more details see [12].
Definition 6.1. We say that M is a monomial OG-module if M =IndGH(N) for some subgroup H of Gand some O-rank one OH-moduleN.
Now consider Hi to be a subgroup of a finite groupGi, for i = 1,2. We would like to study the tensor product of two O-rank one modules.
Lemma 6.2. Let Ni be an O-rank one OHi-module for i = 1,2. The tensor product N1⊗N2 is an O-rank one O[H1×H2]-module.
Proof. It is clear that as anO-module,N1⊗N2 ∼=O ⊗ O ∼=O. For the action, if hi∈Hi andλi ∈ O, fori= 1,2, withhi·ni=λini, we see that the element (h1, h2) in H1×H2 acts on (n1⊗n2)∈N1⊗N2 in such way
(h1, h2)·(n1⊗n2) =λ1n1⊗λ2n2=λ1λ2(n1⊗n2)∈N1⊗N2. We conclude that N1⊗N2 is anO-rank oneO[H1×H2]-module.
The following proposition relates two monomial modules under the exterior ten- sor product.
Proposition 6.3. Let Mi be a monomial OGi-module, for i = 1,2. Then we have M1⊗M2 is a monomial O[G1×G2]-module.
Proof. The assumption that Mi is a monomial OGi-module for i= 1,2 means that there is a subgroup Hi ofGi such that Mi =IndGHi
i(Ni), for someO-rank one OHi-module, fori= 1,2. Using an analogue theory for Lemma 5.2 for modules, we see that
M1⊗M2=IndGH1
1(N1)⊗IndGH2
2(N2) =IndGH1×G2
1×H2(N1⊗N2).
Now Lemma 6.2 completes the proof andM1⊗M2is a monomialO[G1×G2]-module.
A similar theory for endo-permutation modules can be done for monomial mod- ules. Let us introduce the following definition.
Definition 6.4. LetM be anOG-module. We say thatM is an endo-monomial OG-module ifEndO(M) is a monomial OG-module.
Since in this paper we concern to generalize some results in the category of modules to the exterior tensor product, we have the following theorem.
Theorem 6.5. Let Mi be an endo-monomial OGi-module, for i = 1,2. Then we have M1⊗M2 is an endo-monomial O[G1×G2]-module.
Proof. By Definition 6.4, EndO(Mi) is a monomial OGi-module, for i = 1,2.
By Proposition 6.3, EndO(M1)⊗EndO(M2) is a monomialO[G1×G2]-module, for i= 1,2. Now Lemma 3.3 implies thatEndO(M1⊗M2) is a a monomialO[G1×G2]- module. So, M1⊗M2 is an endo-monomial O[G1×G2]-module.
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Ahmad M. Alghamdi
Department of Mathematics, Faculty of Sciences
Umm Alqura University,
P.O. Box 14035, Makkah 21955,Saudi Arabia.
email:[email protected] Makkiah S. Makki
Department of Mathematics
Faculty of Applied Sciences for Girls, Umm Alqura University
P.O. Box 20356 , Makkah 21955, Saudi Arabia.
email:[email protected]
